Calculating quotient algebras of generic embeddings

Many consistency results in set theory involve forcing over a universe V ₀ that contains a large cardinal to get a model V ₁. The original large cardinal embedding is then extended generically using a further forcing by a partial ordering ?. Determining the properties of ? is often the crux of the consistency result. Standard techniques can usually be used to reduce to the case where ? is of the form P(Z)/J for appropriately chosen Z and countably complete ideal J. This paper proves a general algebraic Duality Theorem that exactly characterizes the Boolean algebra P(Z)/J. The Duality Theorem is general enough that it applies even if the original embedding in V ₀ was itself generic. Thus it has as corollaries the theorems of Kakuda, Baumgartner, Laver and others about preservation properties of precipitous and saturated ideals. A corollary is drawn showing that precipitous ideals are indestructible under small proper forcing.